Let be a primitive -th root of unity and look at the following tower of number fields:

where is the field generated by a primitive -th root of unity. This tower of fields has a union . Then the Galois group of over is isomorphic with ; because the Galois group of over is . In order to get an interesting Galois module here, Iwasawa took the ideal class group of , and let be its -torsion part. There are norm mappings when , and so an inverse system. Letting be the inverse limit, we can say that acts on , and it is good to have a description of this action.

The motivation here was undoubtedly that the -torsion in the ideal class group of had already been identified by Kummer as the main obstacle to the direct proof of Fermat's last theorem. What Iwasawa did that was new, was to go 'off to infinity' in a new direction. In fact, is a module over the group ring. This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.

From this beginning, in the 1950s, a good-sized theory has been built up. A basic connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory might be able to move ahead finally from Kummer's century-old results on regular primes.

The main conjecture of Iwasawa theory was formulated as an assertion that two ways of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was eventually proved by Barry Mazur and Andrew Wiles for Q, and for all totally real number fields by Andrew Wiles. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).

More recently, also modeled upon Ribet's method, Chris Skinner and Eric Urban have announced a proof of a main conjecture for GL(2). An easier proof of the Mazur-Wiles theorem can be found by using Euler systems as developed by Kolyvagin (see Washington's book). Other generalizations of the main conjecture proved using the Euler system method have been found by Karl Rubin, amongst others.